Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11366 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #121 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C23×C4×C5 ◁ G | |
634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5)(10,11)(12,17)(13,15)(14,16)(18,19,22,27,23,25,21,26)(20,28,29,32,31,24,30,33);(6,7)(8,9)(10,12,14,13)(11,15,16,17)(18,20,23,31)(19,24,25,28)(21,29,22,30)(26,33,27,32);(7,9)(10,13,14,12)(11,15,16,17)(19,25)(20,29)(21,22)(24,32)(28,33)(30,31)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4354);